googologywikiaorg-20200223-history
Fast-growing hierarchy
A fast-growing hierarchy (FGH) is a certain hierarchy mapping ordinals \(\alpha < \mu\) to functions \(f_\alpha: N \rightarrow N\). For large ordinals \(\alpha\), \(f_\alpha\) grows very rapidly, and the growth rates it achieves are virtually limitless. Due to its simple and clear definition, as well as its origins in professional mathematics, FGH is a popular benchmark for large number functions, alongside BEAF (which is somewhat more esoteric). The functions are usually defined as follows: * \(f_0(n) = n + 1\) * \(f_{\alpha+1}(n) = f^n_\alpha(n)\), where \(f^n\) denotes function iteration * \(f_\alpha(n) = f_{\alphan}(n)\) iff \(\alpha\) is a limit ordinal The general case where \(f_0\) is any increasing function forms a fast iteration hierarchy. \(\alphan\) denotes the \(n\)th term of fundamental sequence assigned to ordinal \(\alpha\). Definitions of \(\alphan\) can vary, giving different fast-growing hierarchies. For \(\alpha \leq \epsilon_0\), the so-called Wainer hierarchy uses the following definition: *\(\omegan = n\) *\(\omega^{\alpha + 1}n = \omega^\alpha n\) *\(\omega^{\alpha}n = \omega^{\alphan}\) iff \(\alpha\) is a limit ordinal *\((\omega^{\alpha_1} + \omega^{\alpha_2} + \cdots + \omega^{\alpha_{k - 1}} + \omega^{\alpha_k})n = \omega^{\alpha_1} + \omega^{\alpha_2} + \cdots + \omega^{\alpha_{k - 1}} + \omega^{\alpha_k}n\) where \(\alpha_1 \geq \alpha_2 \geq \cdots \geq \alpha_{k - 1} \geq \alpha_k\) *\(\epsilon_00 = 0\) (alternatively \(1\)) and \(\epsilon_0+ 1 = \omega^{\epsilon_0n}\) For example, the fundamental sequence for \(\omega^\omega\) is \(1, \omega, \omega^2, \omega^3, \ldots\) The Wainer hierarchy can be extended into the and up to the \(\Gamma_0\). As long as fundamental sequences are defined, it is possible to extend FGH through the recursive ordinals up to \(\omega_1^\text{CK}\) (the ). Whether FGH exists for non-recursive ordinals — and if so, what it means — is an unresolved issue. Approximations Below are some functions in the Wainer hierarchy compared to other googological notations. There are a few things to note: *Relationships denoted \(f_\alpha(n) > g(n)\) hold for sufficiently large \(n\), not necessarily all \(n\). *\(m\) indicates any positive integer. *\(m\alpha\) is here used as a shorthand for \(\alpha \times m = \underbrace{\alpha + \cdots + \alpha}_m\). In standard ordinal arithmetic, \(m \times \alpha = \alpha\) (if \(\alpha\) is transfinite ordinal). *\(^ab\) indicates tetration. *\(\uparrow\) indicates arrow notation. *\(\text{Ack}\) indicates the single-argument Ackermann function \(\text{Ack}(n, n)\). *\(\lbrace \rbrace\) indicates BEAF. Up to \(ε_0\) \begin{eqnarray*} f_0(n) &=& n + 1 \\ f_1(n) &=& f_0^n(n) = ( \cdots ((n + 1) + 1) + \cdots + 1) = n + n = 2n \\ f_2(n) &=& f_1^n(n) = 2(2(\ldots 2(2n))) = 2^n n > 2 \uparrow n \\ f_3(n) &>& 2\uparrow\uparrow n \\ f_4(n) &>& 2\uparrow\uparrow\uparrow n \\ f_m(n) &>& 2\uparrow^{m-1} n \\ f_\omega(n) &>& 2\uparrow^{n-1} n = Ack(n) \\ f_{\omega+1}(n) &>& \lbrace n,n,1,2 \rbrace \\ f_{\omega+2}(n) &>& \lbrace n,n,2,2 \rbrace \\ f_{\omega+m}(n) &>& \lbrace n,n,m,2 \rbrace \\ f_{2\omega}(n) &>& \lbrace n,n,n,2 \rbrace \\ f_{3\omega}(n) &>& \lbrace n,n,n,3 \rbrace \\ f_{m\omega}(n) &>& \lbrace n,n,n,m \rbrace \\ f_{\omega^2}(n) &>& \lbrace n,n,n,n \rbrace \\ f_{\omega^3}(n) &>& \lbrace n,n,n,n,n \rbrace \\ f_{\omega^m}(n) &>& \lbrace n,m+2 (1) 2 \rbrace \\ f_{\omega^{\omega}}(n) &>& \lbrace n,n+2 (1) 2 \rbrace > \lbrace n,n (1) 2 \rbrace \\ f_{\omega^{\omega}+1}(n) &>& \lbrace n,n,2 (1) 2 \rbrace \\ f_{\omega^{\omega}+2}(n) &>& \lbrace n,n,3 (1) 2 \rbrace \\ f_{\omega^{\omega}+m}(n) &>& \lbrace n,n,m+1 (1) 2 \rbrace \\ f_{\omega^{\omega}+\omega}(n) &>& \lbrace n,n,n+1 (1) 2 \rbrace > \lbrace n,n,n (1) 2 \rbrace \\ f_{\omega^{\omega}+\omega+1}(n) &>& \lbrace n,n,1,2 (1) 2 \rbrace \\ f_{\omega^{\omega}+2 \omega}(n) &>& \lbrace n,n,n,2 (1) 2 \rbrace \\ f_{\omega^{\omega}+\omega^2}(n) &>& \lbrace n,n,n,n (1) 2 \rbrace \\ f_ 2}(n) &>& \lbrace n,n (1) 3 \rbrace \\ f_ 3}(n) &>& \lbrace n,n (1) 4 \rbrace \\ f_ m}(n) &>& \lbrace n,n (1) m+1 \rbrace \\ f_{\omega^{\omega+1}}(n) &>& \lbrace n,n (1) n+1 \rbrace > \lbrace n,n (1) n \rbrace \\ f_{\omega^{\omega+2}}(n) &>& \lbrace n,n (1) n,n \rbrace \\ f_{\omega^{\omega+3}}(n) &>& \lbrace n,n,n (1) n,n,n \rbrace \\ f_{\omega^{\omega+m}}(n) &>& \lbrace n,m (1)(1) 2 \rbrace \\ f_{\omega^{2 \omega}}(n) &>& \lbrace n,n (1)(1) 2 \rbrace = \lbrace n,2 (2) 2 \rbrace \\ f_{\omega^{3 \omega}}(n) &>& \lbrace n,n (1)(1)(1) 2 \rbrace = \lbrace n,3 (2) 2 \rbrace \\ f_{\omega^{m \omega}}(n) &>& \lbrace n,m (2) 2 \rbrace \\ f_{\omega^{\omega^2}}(n) &>& \lbrace n,n (2) 2 \rbrace \\ f_{\omega^{\omega^3}}(n) &>& \lbrace n,n (3) 2 \rbrace \\ f_{\omega^{\omega^m}}(n) &>& \lbrace n,n (m) 2 \rbrace \\ f_{\omega^{\omega^\omega}}(n) &>& \lbrace n,n (n) 2 \rbrace = \lbrace n,n (0,1) 2 \rbrace \\ f_{^4{\omega}}(n) &>& \lbrace n,n ((1) 1) 2 \rbrace = n \uparrow\uparrow 3 \&\ n \\ f_{^5{\omega}}(n) &>& \lbrace n,n ((0,1) 1) 2 \rbrace = n \uparrow\uparrow 4 \&\ n \\ f_{^6{\omega}}(n) &>& \lbrace n,n (((1) 1) 1) 2 \rbrace = n \uparrow\uparrow 5 \&\ n \\ f_{^m{\omega}}(n) &>& X \uparrow\uparrow m-1 \&\ n \\ f_{\varepsilon_0}(n) &>& X \uparrow\uparrow n-1 \&\ n \end{eqnarray*} From \(\varepsilon_0\) to \(\Gamma_0\) From here, \(\approx\) indicates a ballpark comparison, meaning that the two functions have similar rates of growth. \begin{eqnarray*} f_{\varepsilon^{\varepsilon_0}_0}(n) &\approx& X \uparrow\uparrow (X+1) \&\ n \\ f_{\varepsilon^{\varepsilon^{\varepsilon_0}_0}_0}(n) &\approx& X \uparrow\uparrow (X+2) \&\ n \\ f_{\varepsilon_1}(n) &\approx& X \uparrow\uparrow (X*2) \&\ n \\ f_{\varepsilon_2}(n) &\approx& X \uparrow\uparrow (X*3) \&\ n \\ f_{\varepsilon_\omega}(n) &\approx& X \uparrow\uparrow (X^2) \&\ n \\ f_{\varepsilon_{2\omega}}(n) &\approx& X \uparrow\uparrow (2*X^2) \&\ n \\ f_{\varepsilon_{\omega^2}}(n) &\approx& X \uparrow\uparrow (X^3) \&\ n \\ f_{\varepsilon_{\omega^\omega}}(n) &\approx& X \uparrow\uparrow (X^X) \&\ n \\ f_{\varepsilon_{\varepsilon_0}}(n) &\approx& X \uparrow\uparrow (X \uparrow\uparrow X) \&\ n \\ f_{ζ_0}(n) &\approx& X \uparrow\uparrow\uparrow X \&\ n \\ f_{η_0}(n) &\approx& X \uparrow^{4} X \&\ n \\ f_{\varphi(m,0)}(n) &\approx& X \uparrow^{m+1} X \&\ n > n \uparrow^{m} \&\ n \\ f_{\varphi(\omega,0)}(n) &\approx& X \uparrow^{n+1} X \&\ n > X \uparrow^{X} \&\ X = \lbrace n,2,1,2 \rbrace \&\ n \\ f_{\varphi(\omega+1,0)}(n) &\approx& \lbrace X,X,X+1 \rbrace \&\ n \\ f_{\varphi(\omega+2,0)}(n) &\approx& \lbrace X,X,X+2 \rbrace \&\ n \\ f_{\varphi(2\omega,0)}(n) &\approx& \lbrace X,X,X*2 \rbrace \&\ n \\ f_{\varphi(3\omega,0)}(n) &\approx& \lbrace X,X,X*3 \rbrace \&\ n \\ f_{\varphi(\omega^2,0)}(n) &\approx& \lbrace X,X,X^2 \rbrace \&\ n \\ f_{\varphi(\omega^3,0)}(n) &\approx& \lbrace X,X,X^3 \rbrace \&\ n \\ f_{\varphi(\omega^\omega,0)}(n) &\approx& \lbrace X,X,X^X \rbrace \&\ n \\ f_{\varphi(\omega^{2\omega},0)}(n) &\approx& \lbrace X,X,X^{X*2} \rbrace \&\ n \\ f_{\varphi(\omega^{3\omega},0)}(n) &\approx& \lbrace X,X,X^{X*3} \rbrace \&\ n \\ f_{\varphi(\omega^{\omega^2},0)}(n) &\approx& \lbrace X,X,X^{X^2} \rbrace \&\ n \\ f_{\varphi(\omega^{\omega^3},0)}(n) &\approx& \lbrace X,X,X^{X^3} \rbrace \&\ n \\ f_{\varphi(\omega^{\omega^\omega},0)}(n) &\approx& \lbrace X,X,X^{X^X} \rbrace \&\ n \\ f_{\varphi(\omega^{\omega^{\omega^\omega}},0)}(n) &\approx& \lbrace X,X,X^{X^{X^X}} \rbrace \&\ n \\ f_{\varphi(\varepsilon_0,0)}(n) &\approx& \lbrace X,X,X \uparrow\uparrow X \rbrace \&\ n \\ f_{\varphi(ζ_0,0)}(n) &\approx& \lbrace X,X,X \uparrow\uparrow\uparrow X \rbrace \&\ n \\ f_{\varphi(η_0,0)}(n) &\approx& \lbrace X,X,X \uparrow^{4} X \rbrace \&\ n \\ f_{\varphi(\varphi(\omega,0),0)}(n) &\approx& \lbrace X,X,X \uparrow^{X} X \rbrace \&\ n = \lbrace X,X, \lbrace X,X,X \rbrace\rbrace \&\ n \\ f_{\varphi(\varphi(\varphi(\omega,0),0),0)}(n) &\approx& \{X,4,2,2\} \&\ n \\ \end{eqnarray*} From \(\Gamma_0\) to \(\vartheta(\Omega^\Omega)\) \begin{eqnarray*} f_{Γ_0}(n) &\approx& \lbrace X,X,1,2 \rbrace \&\ n \\ f_{\varphi(Γ_0,1)}(n) &\approx& \lbrace X,X+1,1,2 \rbrace \&\ n \\ f_{Γ_1}(n) &\approx& \lbrace X,X*2,1,2 \rbrace \&\ n \\ f_{Γ_2}(n) &\approx& \lbrace X,X*3,1,2 \rbrace \&\ n \\ f_{Г_\omega}(n) &\approx& \lbrace X,X^2,1,2 \rbrace \&\ n \\ f_{Г_{Г_0}}(n) &\approx& \lbrace X,\lbrace X,X,1,2 \rbrace,1,2 \rbrace \&\ n \\ f_{\varphi(1,1,0)}(n) &\approx& \lbrace X,X,2,2 \rbrace \&\ n \\ f_{\varphi(1,2,0)}(n) &\approx& \lbrace X,X,3,2 \rbrace \&\ n \\ f_{\varphi(1,\omega,0)}(n) &\approx& \lbrace X,X,X,2 \rbrace \&\ n \\ f_{\varphi(1,\varphi(1,0,0),0)}(n) &\approx& \lbrace X,X,\lbrace X,X,1,2 \rbrace,2 \rbrace \&\ n \\ f_{\varphi(2,0,0)}(n) &\approx& \lbrace X,X,1,3 \rbrace \&\ n \\ f_{\varphi(3,0,0)}(n) &\approx& \lbrace X,X,1,4 \rbrace \&\ n \\ f_{\varphi(\omega,0,0)}(n) &\approx& \lbrace X,X,1,X \rbrace \&\ n \\ f_{\varphi(\varphi(1,0,0),0,0)}(n) &\approx& \lbrace X,X,1,\lbrace X,X,1,2 \rbrace\rbrace \&\ n \\ f_{\varphi(1,0,0,0)}(n) &\approx& \lbrace X,X,1,1,2 \rbrace \&\ n \\ f_{\varphi(2,0,0,0)}(n) &\approx& \lbrace X,X,1,1,3 \rbrace \&\ n \\ f_{\varphi(1,0,0,0,0)}(n) &\approx& \lbrace X,X,1,1,1,2 \rbrace \&\ n \\ f_{\vartheta(\Omega^\omega)}(n) &\approx& \lbrace X,X (1) 2 \rbrace \&\ n \\ f_{\vartheta(\Omega^{\omega+1})}(n) &\approx& \lbrace X,X (1) X \rbrace \&\ n \\ f_{\vartheta(\Omega^{\omega+2})}(n) &\approx& \lbrace X,X (1) X,X \rbrace \&\ n \\ f_{\vartheta(\Omega^{2\omega})}(n) &\approx& \lbrace X,X (1)(1) 2 \rbrace \&\ n \\ f_{\vartheta(\Omega^{3\omega})}(n) &\approx& \lbrace X,X (1)(1)(1) 2 \rbrace \&\ n \\ f_{\vartheta(\Omega^{\omega^2})}(n) &\approx& \lbrace X,X (2) 2 \rbrace \&\ n \\ f_{\vartheta(\Omega^{\omega^3})}(n) &\approx& \lbrace X,X (3) 2 \rbrace \&\ n \\ f_{\vartheta(\Omega^{\omega^\omega})}(n) &\approx& \lbrace X,X (0,1) 2 \rbrace \&\ n \\ f_{\vartheta(\Omega^{\omega^{\omega^\omega}})}(n) &\approx& \lbrace X,X ((1) 1) 2 \rbrace \&\ n \\ f_{\vartheta(\Omega^{\vartheta(1)})}(n) &\approx& X \uparrow\uparrow X \&\ X \&\ n \\ f_{\vartheta(\Omega^{\vartheta(2)})}(n) &\approx& X \uparrow\uparrow\uparrow X \&\ X \&\ n \\ f_{\vartheta(\Omega^{\vartheta(3)})}(n) &\approx& X \uparrow^{4} X \&\ X \&\ n \\ f_{\vartheta(\Omega^{\vartheta(\omega)})}(n) &\approx& X \uparrow^{X} X \&\ X \&\ n \\ f_{\vartheta(\Omega^{\vartheta(\Omega)})}(n) &\approx& \lbrace X,X,1,2 \rbrace \&\ X \&\ n \\ f_{\vartheta(\Omega^{\vartheta(\Omega^\omega)})}(n) &\approx& \lbrace X,X (1) 2 \rbrace \&\ X \&\ n \\ f_{\vartheta(\Omega^{\vartheta(\Omega^{\vartheta(1)})})}(n) &\approx& X \uparrow\uparrow X \&\ X \&\ X \&\ n \\ f_{\vartheta(\Omega^{\vartheta(\Omega^{\vartheta(\Omega^{\vartheta(1)})})})}(n) &\approx& X \uparrow\uparrow X \&\ X \&\ X \&\ X \&\ n \\ \end{eqnarray*} Beyond \(\vartheta(\Omega^\Omega)\) \begin{eqnarray*} f_{\vartheta(\Omega^\Omega)}(n) &\approx& \lbrace n,n / 2 \rbrace \\ f_{\vartheta(\Omega^\Omega)+1}(n) &\approx& \lbrace n,n,2 / 2 \rbrace \\ f_{\vartheta(\Omega^\Omega)+2}(n) &\approx& \lbrace n,n,3 / 2 \rbrace \\ f_{\vartheta(\Omega^\Omega)+\omega}(n) &\approx& \lbrace n,n,n / 2 \rbrace \\ f_{\vartheta(\Omega^\Omega)+\omega^2}(n) &\approx& \lbrace n,n,n,n / 2 \rbrace \\ f_{\vartheta(\Omega^\Omega)+\omega^3}(n) &\approx& \lbrace n,n,n,n,n / 2 \rbrace \\ f_{\vartheta(\Omega^\Omega)+\omega^\omega}(n) &\approx& \lbrace n,n (1) 2 / 2 \rbrace \\ f_{\vartheta(\Omega^\Omega)+\omega^{\omega^\omega}}(n) &\approx& \lbrace n,n (0,1) 2 / 2 \rbrace \\ f_{\vartheta(\Omega^\Omega)+\varepsilon_0}(n) &\approx& \lbrace X \uparrow\uparrow X \&\ n / 2 \rbrace \\ f_{\vartheta(\Omega^\Omega)+\zeta_0}(n) &\approx& \lbrace X \uparrow\uparrow\uparrow X \&\ n / 2 \rbrace \\ f_{\vartheta(\Omega^\Omega)+\eta_0}(n) &\approx& \lbrace X \uparrow\uparrow\uparrow\uparrow X \&\ n / 2 \rbrace \\ f_{\vartheta(\Omega^\Omega)+\Gamma_0}(n) &\approx& \{X,X,1,2\} \&\ n / 2 \rbrace \\ f_{2 \vartheta(\Omega^\Omega)}(n) &\approx& \lbrace n,n / 3 \rbrace \\ f_{3 \vartheta(\Omega^\Omega)}(n) &\approx& \lbrace n,n / 4 \rbrace \\ f_{\omega \vartheta(\Omega^\Omega)}(n) &\approx& \lbrace n,n / n \rbrace \\ f_{\omega +1 \vartheta(\Omega^\Omega)}(n) &\approx& \lbrace n,n / 1,2 \rbrace \\ f_{\omega +2 \vartheta(\Omega^\Omega)}(n) &\approx& \lbrace n,n / 2,2 \rbrace \\ f_{2 \omega \vartheta(\Omega^\Omega)}(n) &\approx& \lbrace n,n / n,2 \rbrace \\ f_{3 \omega \vartheta(\Omega^\Omega)}(n) &\approx& \lbrace n,n / n,3 \rbrace \\ f_{\omega^2 \vartheta(\Omega^\Omega)}(n) &\approx& \lbrace n,n / n,n \rbrace \\ f_{\omega^3 \vartheta(\Omega^\Omega)}(n) &\approx& \lbrace n,n / n,n,n \rbrace \\ f_{\omega^\omega \vartheta(\Omega^\Omega)}(n) &\approx& \lbrace n,n / 1 (1) 2 \rbrace \\ f_{\omega^{\omega^\omega} \vartheta(\Omega^\Omega)}(n) &\approx& \lbrace n,n / 1 (0,1) 2 \rbrace \\ f_{\varepsilon_0 \vartheta(\Omega^\Omega)}(n) &\approx& \lbrace n,n / X \uparrow\uparrow X \&\ n \rbrace \\ f_{\vartheta(\Omega^\Omega)^{2}}(n) &\approx& \lbrace n,n / 1 / 2 \rbrace \\ f_{\vartheta(\Omega^\Omega)^{3}}(n) &\approx& \lbrace n,n / 1 / 1 / 2 \rbrace \\ f_{\vartheta(\Omega^\Omega)^{\omega}}(n) &\approx& \lbrace n,n (/1) 2 \rbrace \\ f_{\vartheta(\Omega^\Omega)^{\omega^2}}(n) &\approx& \lbrace n,n (/2) 2 \rbrace \\ f_{\vartheta(\Omega^\Omega)^{\omega^3}}(n) &\approx& \lbrace n,n (/3) 2 \rbrace \\ f_{\vartheta(\Omega^\Omega)^{\omega^\omega}}(n) &\approx& \lbrace n,n (/0,1) 2 \rbrace \\ f_{\vartheta(\Omega^\Omega)^{\varepsilon_0}}(n) &\approx& X \uparrow\uparrow X \&\ \&\ n \\ f_{\vartheta(\Omega^\Omega)^{\Gamma_0}}(n) &\approx& \{X,X,1,2\} \&\ \&\ n \\ f_{\vartheta(\Omega^\Omega)^{\vartheta(\Omega^\Omega)}}(n) &\approx& \lbrace X,X / 2 \rbrace \&\ \&\ n \\ f_{\varepsilon_{\vartheta(\Omega^\Omega)+1}}(n) &\approx& \lbrace n,n // 2 \rbrace = \{L,2\}_{n,n}\\ f_{\varepsilon_{\vartheta(\Omega^\Omega)+1}^\varepsilon}(n) &\approx& X \uparrow\uparrow X \&\ \&\ \&\ n \\ f_{\varepsilon_{\vartheta(\Omega^\Omega)+2}}(n) &\approx& \lbrace n,n /// 2 \rbrace = \{L,3\}_{n,n}\\ f_{\varepsilon_{\vartheta(\Omega^\Omega)+3}}(n) &\approx& \lbrace n,n //// 2 \rbrace = \{L,4\}_{n,n}\\ f_{\varepsilon_{\vartheta(\Omega^\Omega)+\omega}}(n) &\approx& \lbrace n,n (1)/ 2 \rbrace = \{L,X\}_{n,n}\\ f_{\varepsilon_{\vartheta(\Omega^\Omega)+\omega^2}}(n) &\approx& \lbrace n,n (2)/ 2 \rbrace = \{L,X^2\}_{n,n}\\ f_{\varepsilon_{\vartheta(\Omega^\Omega)+\omega^3}}(n) &\approx& \lbrace n,n (3)/ 2 \rbrace = \{L,X^3\}_{n,n}\\ f_{\varepsilon_{\vartheta(\Omega^\Omega)+\omega^\omega}}(n) &\approx& \lbrace n,n (0,1)/ 2 \rbrace = \{L,X^X\}_{n,n}\\ f_{\varepsilon_{\vartheta(\Omega^\Omega)+\omega^{\omega^\omega}}}(n) &\approx& \lbrace n,n ((1)1)/ 2 \rbrace = \{L,X^{X^X}\}_{n,n}\\ f_{\varepsilon_{\vartheta(\Omega^\Omega)+\varepsilon_0}}(n) &\approx& \{L,\{X,X,2\}\}_{n,n}\\ f_{\varepsilon_{\vartheta(\Omega^\Omega)+\zeta_0}}(n) &\approx& \{L,\{X,X,3\}\}_{n,n}\\ f_{\varepsilon_{\vartheta(\Omega^\Omega)+\eta_0}}(n) &\approx& \{L,\{X,X,4\}\}_{n,n}\\ f_{\varepsilon_{\vartheta(\Omega^\Omega)+\varphi(\omega,0)}}(n) &\approx& \{L,\{X,X,X\}\}_{n,n}\\ f_{\varepsilon_{\vartheta(\Omega^\Omega)+\varphi(\varphi(\omega,0),0)}}(n) &\approx& \{L,\{X,X,\{X,X,X\}\}\}_{n,n}\\ f_{\varepsilon_{\vartheta(\Omega^\Omega)+\Gamma_0}}(n) &\approx& \{L,\{X,X,1,2\}\}_{n,n}\\ f_{\varepsilon_{\vartheta(\Omega^\Omega)+\varphi(2,0,0)}}(n) &\approx& \{L,\{X,X,1,3\}\}_{n,n}\\ f_{\varepsilon_{\vartheta(\Omega^\Omega)+\varphi(\omega,0,0)}}(n) &\approx& \{L,\{X,X,1,X\}\}_{n,n}\\ f_{\varepsilon_{\vartheta(\Omega^\Omega)+\varphi(1,0,0,0)}}(n) &\approx& \{L,\{X,X,1,1,2\}\}_{n,n}\\ f_{\varepsilon_{\vartheta(\Omega^\Omega)+\vartheta(\Omega^\omega)}}(n) &\approx& \{L,\{X,X (1) 2\}\}_{n,n}\\ f_{\varepsilon_{\vartheta(\Omega^\Omega)+\vartheta(\Omega^{\omega^2})}}(n) &\approx& \{L,\{X,X (2) 2\}\}_{n,n}\\ f_{\varepsilon_{\vartheta(\Omega^\Omega)+\vartheta(\Omega^{\omega^\omega})}}(n) &\approx& \{L,\{X,X (0,1) 2\}\}_{n,n}\\ f_{\varepsilon_{\vartheta(\Omega^\Omega)+\vartheta(\Omega^{\vartheta(1)})}}(n) &\approx& \{L,X \uparrow\uparrow X \& X \}_{n,n}\\ f_{\varepsilon_{2 \vartheta(\Omega^\Omega)}}(n) &\approx& \{L,L\}_{n,n}\\ f_{\varepsilon_{3 \vartheta(\Omega^\Omega)}}(n) &\approx& \{L,L,L\}_{n,n}\\ f_{\varepsilon_{\omega\vartheta(\Omega^\Omega)}}(n) &\approx& \{L,X (1) 2\}_{n,n}\\ f_{\varepsilon_{\vartheta(\Omega^\Omega)^2}}(n) &\approx& \{L,L (1) 2\}_{n,n}\\ f_{\varepsilon_{\vartheta(\Omega^\Omega)^3}}(n) &\approx& \{L,L,L (1) 2\}_{n,n}\\ f_{\varepsilon_{\vartheta(\Omega^\Omega)^\omega}}(n) &\approx& \{L,L (1) 3\}_{n,n}\\ f_{\varepsilon_{\vartheta(\Omega^\Omega)^{2\omega}}}(n) &\approx& \{L,L (1) 4\}_{n,n}\\ \end{eqnarray*} Non-recursive ordinals It is possible to define the fast-growing hierarchy for all recursive ordinals, and even for nonrecursive ordinals. However, the definitions will necessarily be nonrecursive, making analysis far more complicated. For example, it is a nontrivial question whether \(F_{\omega^\text{CK}_1}(n)\) outgrows all computable functions, and therefore is comparable in growth rate to \(\Sigma(n)\). The smallest non-recursive ordinal is \(\omega^\text{CK}_1\), the . BEAF, being a computable function, is completely exhausted by now. In order, the functions are Rado's sigma function, its higher-order cousins, Rayo's function, and the xi function. Here \(\alpha\) is the first ordinal such that \(\omega^\text{CK}_{\alpha} = \alpha\). \begin{eqnarray*} f_{\omega^\text{CK}_1}(n) &\approx& \Sigma(n) \\ f_{\omega^\text{CK}_2}(n) &\approx& \Sigma_2(n) \\ f_{\omega^\text{CK}_3}(n) &\approx& \Sigma_3(n) \\ f_{\omega^\text{CK}_m}(n) &\approx& \Sigma_m(n) \\ f_{\alpha}(n) &\approx& \Xi(n) \\ \end{eqnarray*} See also *List of googological functions Category:Hierarchies Category:Functions